Your camera does a lot of processing of the image before recording it on the memory card. Colour, brightness, contrast, noise, sharpness, etc., are all adjusted as the camera sees fit. When it is saved on the card as a JPEG, the image is compressed, losing some of its quality in the process.

If your camera can shoot RAW, the in-camera processing is by-passed, and the image is stored on your card exactly as it appeared on the sensor. Also, there's no compression, so the image is the best quality your camera can produce.

You then post-process the image in a RAW converter, where you can adjust the characteristics of the image as you please, before saving as a JPEG, TIFF, etc.

White balance is a messy at times. For example, say you're shooting a person who is on a theater stage. And the stage lighting is yellow. The yellow light shining on a Caucasian actress makes her skin look yellow and sick. This type of effect is described as the white balance being off. If you're not shooting in RAW, then you're likely to do some in camera changes to minimize the funky skin color and you're probably going to get to do some Photoshop changes as well.

I've learned that fixing this skin color issue has become much easier when shooting in RAW. I'm not saying you can't fix it in Photoshop. But I've been amazed at how much easier it is to fix in RAW. Not only does it fix the skin color, it fixes the other colors as well while you're fixing the skin color issue.

The program I use to do this is Lightroom II. I'm a recent convert of Lightroom. IMHO, it's a headache to learn because it truly works differently than other photo editors. But once you get past the initial hurdle, it becomes pretty straight forward.

The other advantage of Lightroom is that you can "batch" process. Say I shot 100 photos of a theater presentation. I can adjust the white balance for one, and then chose to clone those changes to all the other shots that were taken under similar lighting. Think that doesn't save you time and effort?!

Shooting in RAW is great if you've got the time and storage capacity, and you're short on opportunity.

For all the reasons neilcrichton and FaithfulPastor mentioned, and a few more no doubt, RAW is a good way to go. But shooting in RAW requires you to do some post processing, whereas JPEG images can be good to go right out of the camera. Also, since RAW files are larger than JPEG files, the rate of continuous shooting is slower, and transfering images from camera to computer takes longer, and they take up more space on the flash memory card and on your computer.

But if you've only got one chance to get the shot right, shooting RAW will let you correct for some difficiencies in the original image that would be tedious if not impossible with a JPEG image.

Some cameras also offer RAW + JPG. In this mode, the camera is storing both the unprocessed RAW and the processed JPG, and you have less room on your memory card for images. There is also an advantage, whereby if you want the camera processed JPG and the unprocessed RAW stored on your PC for later reference or processing. It is just another option, that works for some and for others offers nothing.

Raw mode requires at least one extra step in order to get an image you can view on screen or print. A Raw converter must be used to turn the raw file into a viewable image file, such as jpg, or tiff. Raw converters give you many options on how to process your picture, including noise reduction, sharpness, WB, and exposure. Some will allow cropping, and tone mapping as well. In many, if not most, cases, this precludes the need for using an editor, such as Photoshop. It does take more time on your part though.
For casual use, such as family gatherings and when I'm just out and about with my camera, I usually have it set to jpeg, since it is easy enough to change if a special shot appears. Some cameras now have dedicated buttons for switching to raw in circumstances that call for it.